Rasch model analysis helps create measurement scales based on test items (or survey items. For simplicity, let's pretend our goal is to create test scores.

To create test scores, Rasch model utilizes three sources of information: a) who the respondents were (coded as a series of 0-1 dummy variables) and b) what the items were (coded as a series of dummy variables), and c) what the response was (Correct vs. Incorrect). The following summarizes the Rasch equation. Notice that this is a logistic regression where the outcome is binary.

Log(P/1-P)= b1*john + b2*Edward + b3*Mary …. + g1*item1 + g2*item2 + g3*item3

P stands for a probability of a person getting a correct answer for a test item. Person measures, b1, b2, b3. ... are the products of this analysis and they correspond to each respondent's score.

The dataset would look like this. This is a hypothetical dataset.

Correct (1) or wrong (0) | John | Edward | Mary | Item1 | Item2 | Item3 |

0 | 1 | 0 | 0 | 1 | 0 | 0 |

0 | 1 | 0 | 0 | 0 | 1 | 0 |

1 | 1 | 0 | 0 | 0 | 0 | 1 |

1 | 0 | 1 | 0 | 1 | 0 | 0 |

0 | 0 | 1 | 0 | 0 | 1 | 0 |

1 | 0 | 1 | 0 | 0 | 0 | 1 |

1 | 0 | 0 | 1 | 1 | 0 | 0 |

0 | 0 | 0 | 1 | 0 | 1 | 0 |

1 | 0 | 0 | 1 | 0 | 0 | 1 |

Again:

Log(P/1-P)= b1*john + b2*Edward + b3*Mary …. + g1*item1 + g2*item2 + g3*item3

- b1, b2, and b3 are person measures.
- g1, g2, and g3 are item measures (they measure how easy each items are). If you multiple these with -1, they become item difficulty measures.

Mike Lanacre's Winsteps is a good software for Rasch model analysis. I learned Rasch model from Winstep author Mike Linacre and Rasch guru Benjamin Wright in 1995.

Special topics

- Advantage of Rasch model scores
- QC
- Vertical equated scores
- Anchored item analysis
- Passing of Dr. Ben Wright: https://news.uchicago.edu/article/2015/12/15/benjamin-wright-renowned-psychometrician-1926-2015